Lyapunov Instabilities of Extended Systems

One of the most successful theories in modern science is statistical mechanics, which allows us to understand the macroscopic (thermodynamic) properties of matter from a statistical analysis of the microscopic (mechanical) behavior of the constituent particles. In spite of this, using certain probabilistic assumptions such as Boltzmann’s

Stosszahlansatz

causes the lack of a firm foundation of this theory, especially for non-equilibrium statistical mechanics. Fortunately, the concept of chaotic dynamics developed in the 20th century [1] is a good candidate for accounting for these difficulties. Instead of the probabilistic assumptions, the dynamical instability of trajectories can make available the necessary fast loss of time correlations, ergodicity, mixing and other dynamical randomness [2]. It is generally expected that dynamical instability is at the basis of macroscopic transport phenomena and that one can find certain connections between them. Some beautiful theories in this direction were already developed in the past decade. Examples are the escape-rate formalism by Gaspard and Nicolis [3, 4] and the Gaussian thermostat method by Nosé, Hoover, Evans, Morriss and others [5, 6], where the Lyapunov exponents were related to certain transport coefficients.